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### The Helly bound for singular sums

Robert E. Jamison
2002 Discrete Mathematics
In this paper we will establish a bound for the number of sums which can be generated by a clique in the singularity graph of Zn, the ring of integers modulo n.  ...  When n has at least three prime factors, there are always cliques based on Helly families of sets which realize n − (n) sums, where denotes the Euler totient function.  ...  The author is grateful to the referees for suggesting that it could be extended.  ...

### PROOF OF A CONJECTURE OF STEINHAUS

H. Helson
1954 Proceedings of the National Academy of Sciences of the United States of America
By the Helly theorem there is a function ,(x) of bounded variation and a subsequence of j such that  ...  Steinhaus has conjectured' the following theorem: Suppose a trigonometric series -CD is given with the property that the partial sums N E aneinx -N are non-negative for all x and all sufficiently large  ...

### On a topological fractional Helly theorem [article]

Stephan Hell
2005 arXiv   pre-print
We prove a new fractional Helly theorem for families of sets obeying topological conditions.  ...  This implies fractional Helly number k+1 for families F. Moreover, we obtain a topological (p,q)-theorem.  ...  Acknowledgement I thank Carsten Schultz for suggesting the use of the spectral sequence argument, and my advisor Günter M. Ziegler for helpful discussions. Many thanks also to Jiří  ...

### Bounding Helly Numbers via Betti Numbers [chapter]

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner
2017 A Journey Through Discrete Mathematics
Here βi denotes the reduced Z 2 -Betti numbers (with singular homology).  ...  If F is a finite family of subsets of R d such that βi ( G) ≤ b for any G F and every 0 ≤ i ≤ d/2 − 1 then F has Helly number at most h(b, d).  ...  We would like to express our immense gratitude to Jiří Matoušek, not only for raising the problem addressed in the present paper and valuable discussions about it, but, much more generally, for the privilege  ...

### Page 239 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 6, Issue 2 [page]

1955 American Mathematical Society. Proceedings of the American Mathematical Society
,> bice-t- k=—oo As p tends to infinity through the sequence of values pi, pz, - - - the sum tends to d, = , bic g_x. k=0 But the total variation of the measures is uniformly bounded, and so by the Helly  ...  The first lemma asserts directly that yp is singular. LemMA. The singular measure yp is also absolutely continuous. If K is a constant larger than any | },|, then & @ ld.]  ...

### Bounding Helly numbers via Betti numbers [article]

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner
2016 arXiv   pre-print
Here β̃_i denotes the reduced Z_2-Betti numbers (with singular homology).  ...  We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem.  ...  , for the privilege of having known him, as our teacher, mentor, collaborator, and friend.  ...

### Author index

2002 Discrete Mathematics
., The Helly bound for singular sums (1-3) 117-133 Faudree, R.J., R.J. Gould, M.S. Jacobson and L.L.  ...  ., Hall's condition for list-coloring, and the Hall parameters: recent developments (1-3) 135-147 K .undgen, A., Minimum average distance subsets in the hamming cube  ...

### Page 247 of Mathematical Reviews Vol. 4, Issue 9 [page]

1943 Mathematical Reviews
[MF 7334] Let S be a partially ordered vector space in which the existence of an upper bound for a monotone increasing sequence implies the existence of a least upper bound.  ...  It is shown that w(Z) compactness of order X, implies the extended Helly property of order X,, and that, if the extended Helly property holds for all orders, then the space is w(Z) complete.  ...

### Bounding Radon's number via Betti numbers [article]

Zuzana Patáková
2019 arXiv   pre-print
Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets  ...  Then if F is a finite family of sets in X such that β_i( G; Z_2) is at most b for all i=0,1,..., k and G⊆ F, then the Radon's number of F is bounded in terms of b and X.  ...  Finally, many thanks to Natan Rubin for several discussions at the very beginning of the project.  ...

### Quantitative combinatorial geometry for concave functions [article]

Sherry Sarkar, Alexander Xue, Pablo Soberón
2020 arXiv   pre-print
Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single H-convex set.  ...  Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure.  ...  The authors would like to thank Emo Welzl for his helpful comments and for pointing out the algorithmic applications of our results in the framework of LP-type problems.  ...

### Development of Hankel Singular-Hypergraph Feature Extraction Technique for Acoustic Partial Discharge Pattern Classification

Suganya Govindarajan, Venkateshwar Ragavan, Ayman El-Hag, Kannan Krithivasan, Jayalalitha Subbaiah
2021 Energies
Recent research posits that features extracted by singular value decomposition (SVD) can exhibit the natural characteristics and energy contained in the signal.  ...  The algorithm is tested for various measurement conditions that include the influences of various PD locations and oil temperatures.  ...  Acknowledgments: The authors thank the Management of SASTRA University and Tata Realty-IT City-SASTRA Srinivasa Ramanujan Research cell.  ...

### On a Theorem of Szego

Henry Helson
1955 Proceedings of the American Mathematical Society
Jfc-«-00 As p tends to infinity through the sequence of values pi, p2, • • • the sum tends to 00 dq = £ bkCq-k. fc-0 But the total variation of the measures is uniformly bounded, and so by the Helly theorem  ...  Since p, is a singular measure, for almost every point x this variation is o(e) as e tends to zero.  ...

### On a theorem of Szegö

Henry Helson
1955 Proceedings of the American Mathematical Society
Jfc-«-00 As p tends to infinity through the sequence of values pi, p2, • • • the sum tends to 00 dq = £ bkCq-k. fc-0 But the total variation of the measures is uniformly bounded, and so by the Helly theorem  ...  Since p, is a singular measure, for almost every point x this variation is o(e) as e tends to zero.  ...

### Bounding Radon Number via Betti Numbers

Zuzana Patáková, Danny Z. Chen, Sergio Cabello
2020 International Symposium on Computational Geometry
Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets  ...  to k, then the Radon number of F is bounded in terms of b and X.  ...  From these consequences only the fact that for sets in R d bounded T C d/2 implies bounded Helly number has been shown earlier  .  ...

### Helly numbers of acyclic families

Éric Colin de Verdière, Grégory Ginot, Xavier Goaoc